I'm just gonna pop this solution here in case anybody else has had the same trouble as me, because this 'cheev (and a few others) seems to be a little glitchy. If you're sure you've killed 300 Stormtroopers and it hasn't unlocked, here's a sure fire way to get it (you need a dummy account and an extra controller):
1. Clear your cache.
2. Start up LEGO, don't update and stay offline. Load up your game.
3. Sign out of your account, sign in with the dummy account. It won't put you back on the main menu.
4. Sign back in as your normal account with the second controller and get the achievement(s) as the second player and it should unlock.
This worked for me for this achievement, along with the hat and the Tie-fighters one, all of which seemed to be glitched. I assume it'll work for every other achievement apart from the Story-related ones.
EDIT: Just checked with a couple of other achievements I needed, and you don't need to stay offline. So you may not even need to clear your cache, just make sure that you're getting them as the second player and you're set. :)